= Solution
Let $b\simeq2l_p$ be the <Kuhn length>, $N=L/b$ the number of statistical segments, and $v$ their effective <excluded volume>. We first assume a good-solvent, strongly confined <self-avoiding walk>, rather than an <ideal chain>. A confinement blob contains $g$ links and has diameter $D\sim bg^{3/5}$ in the simple $v\sim b^3$ scaling model. Therefore $g\sim(D/b)^{5/3}$. Distant blobs cannot freely pass through one another in a narrow tube, so the $N/g$ blobs form an axial string. The <de Gennes confinement scaling> is consequently
$$
\boxed{X\sim\frac NgD\sim L\left(\frac bD\right)^{2/3}\sim L\left(\frac{l_p}{D}\right)^{2/3}.}
$$
The <free energy> is of order $k_BT$ per blob, not the <energy> of stretching a straight elastic rod.
The role of solvent and chain thickness can be made explicit by a <Flory approximation>: an axial extension $X$ costs an entropic term $k_BTX^2/(Nb^2)$, while the confined number density gives an interaction term $k_BTvN^2/(XD^2)$. Minimizing their sum gives $X^3\sim vN^3b^2/D^2$, hence
$$
\boxed{X\sim L\left(\frac{v}{bD^2}\right)^{1/3}.}
$$
For a thin statistical segment of effective diameter $d$, $v\sim b^2d$ instead gives $X\sim L(bd/D^2)^{1/3}$. Thus the simpler answer implicitly fixes the <excluded volume>; the <DNA> thickness and ionic environment can matter quantitatively.
Both blob arguments require $b\ll D\ll R_{\rm free}$ and $N/g\gg1$, where $R_{\rm free}\sim bN^{3/5}$ in the simple good-solvent model. The printed comparison with $L$ alone does not ensure confinement relative to the unconfined coil. If $D\gg R_{\rm free}$, the tube barely perturbs the coil. An <ideal chain> without <excluded volume> has axial spread of order $\sqrt{Lb}$ and does not acquire the extensive extension of a nonoverlapping blob string. These are distinct physical regimes, not changes to a numerical prefactor.
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