= Solution
The given $\gamma$ is a <simple closed curve>. Assume it is essential; for a contractible class both infimal lengths are already zero. Write $\ell_n=\ell_{X_n}(\gamma)$. The <collar lemma> supplies an embedded <annulus> with coordinates $r\in(-w_n,w_n)$ and $t\in\mathbb R/\mathbb Z$, and metric
$$
ds^2=dr^2+\ell_n^2\cosh^2r\,dt^2,\qquad
\sinh w_n\,\sinh(\ell_n/2)=1.
$$
Set $y(r)=\ell_n^{-1}\int_0^r\operatorname{sech}s\,ds$. This changes the metric to a positive scalar multiple of $dt^2+dy^2$, so the <conformal modulus of an annulus> is
$$
M_n=\frac2{\ell_n}\int_0^{w_n}\operatorname{sech}r\,dr
=\frac2{\ell_n}\arctan\!\left(\frac1{\sinh(\ell_n/2)}\right)
\sim\frac\pi{\ell_n}\longrightarrow\infty.
$$
The <extremal length> of its core curves is $1/M_n$. Allowing all curves <homotopic> to $\gamma$ in $X_n$ can only decrease infimal lengths, while the area of a metric on all of $X_n$ is at least its area on the collar. Thus $\lambda(\gamma,X_n)\leq1/M_n$. Use the particular <conformal metric> $|q_n|^{1/2}$, whose <area of a quadratic differential> is one:
$$
\boxed{L(\gamma,|q_n|^{1/2})^2
\leq\lambda(\gamma,X_n)\leq\frac1{M_n}\longrightarrow0.}
$$
This proves the required implication uniformly over all the area-one <holomorphic quadratic differentials> on these surfaces.
\b[The converse is false.] Here is an explicit <slit connected sum of translation tori>. Start with square flat copies of a <torus> $T_1=\mathbb C/(\mathbb Z+i\mathbb Z)$ and $T_\delta=\mathbb C/(\delta\mathbb Z+i\delta\mathbb Z)$, where $\delta\to0$. Cut a horizontal slit of physical length $\delta^2$ in each, centred in an interior coordinate disk, and cross-glue the banks by translation. The resulting surface has <genus> two. Its two slit endpoints have <cone angle> $4\pi$, so the locally defined $dz$ extends to a <holomorphic one-form> $\omega_\delta$ with two simple zeros. Its flat area is $1+\delta^2$. Set
$$
q_\delta=\frac{\omega_\delta^2}{1+\delta^2},
$$
which is a <holomorphic quadratic differential> of area one. Let $\gamma$ be a horizontal generator in the small <torus>, taken away from the slit and fixed by the marking of this small handle. Then
$$
L(\gamma,|q_\delta|^{1/2})\leq\frac{\delta}{\sqrt{1+\delta^2}}\longrightarrow0.
$$
To verify that its hyperbolic length does not tend to zero, construct a uniform lower bound on <extremal length>. On the unit square <torus> choose a disk $D$ about the eventual slit centre and a <smooth cutoff function> $\chi$ equal to one on a smaller disk and supported in $D$. Let $x$ denote a local real coordinate on $D$; in the first term below, $dx$ is the globally defined torus one-form, while $\chi x$ is extended by zero outside $D$. The real <closed differential form>
$$
\alpha=dx-d(\chi x)
$$
is globally defined, vanishes on the smaller disk, and has period one on the horizontal generator. Pull it to $T_\delta$ by the rescaling map $z\mapsto z/\delta$, and extend it by zero across the slit and over the other <torus>. For sufficiently small $\delta$, the slit is inside the region where the form vanishes. Hence this extension $\alpha_\delta$ is a <smooth> <closed differential form> on the <connected> sum, with $\int_\gamma\alpha_\delta=1$.
Define a nonnegative <conformal metric> density by the pointwise norm of $\alpha_\delta$ relative to the flat metric. Two-dimensional scale invariance gives
$$
A_\rho=\int|\alpha_\delta|^2\,dA=\int_{T_1}|\alpha|^2\,dA=:C<\infty,
$$
independently of $\delta$. For every representative $\widetilde\gamma$ <homotopic> to $\gamma$,
$$
\int_{\widetilde\gamma}\rho\,ds
\geq\left|\int_{\widetilde\gamma}\alpha_\delta\right|=1,
$$
because the period of a <closed differential form> is unchanged by homotopy. This is the <extremal length lower bound from a closed one-form>; therefore $\lambda(\gamma,X_\delta)\geq1/C>0$. If $\ell_{X_\delta}(\gamma)$ tended to zero along any subsequence, the collar estimate would force $\lambda(\gamma,X_\delta)\to0$, a contradiction. In fact its hyperbolic lengths are uniformly bounded away from zero. Thus
$$
\boxed{L(\gamma,|q_\delta|^{1/2})\to0
\quad\text{while}\quad
\inf_{\delta\text{ small}}\ell_{X_\delta}(\gamma)>0.}
$$
For every fixed $g>2$, replace $T_1$ by the area-one <translation surface> $X_{2g-2}$ constructed in the polygon argument; its <genus> is $g-1$. Cut its slit inside a nonsingular <flat coordinate> disk. The new connected sum has <genus> $g$, and the area normalization, small-handle length bound and closed-one-form energy argument are unchanged. Thus the converse fails at every fixed <genus> $g\geq2$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-132-shrinking-handle.png]
{title=A small translation torus joined by equal slits; its generator is flat-short while retaining a positive extremal-length bound}
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