= Solution
Write $H(R)$ and $V(R)$ for the horizontal and vertical open-crossing <increasing events> of a rectangle in the rotated <square lattice>. At parameter $1/2$, <planar duality for rectangle crossings> says that failure of $H$ is exactly a closed dual vertical crossing. Quarter-turn symmetry and the boundary convention of the rotated rectangles identify the two alternatives on a square. Thus its crossing <probability> is $1/2$; a rectangle obtained by a harmless one-step boundary adjustment has crossing <probability> at least $1/2$. This is the finite-domain <exact self-dual rectangle crossing probability>, not an assumption about infinite <percolation clusters>.
We first prove a <RSW reflection extension lemma>. In coordinate units adapted to the rotated drawing, take $S=[0,n]^2$, $T=[-n,n]^2$, and $R=[-n,2n]\times[-n,n]$. Reveal the rightmost open vertical crossing $\Gamma$ of $S$, when one exists, using only its <edges> and the region to its right. Reflect $\Gamma$ across the horizontal axis. The part $D$ of $T$ to the left of these two paths remains unexposed and has the independent parameter-$1/2$ law. A horizontal crossing of $T$ must reach their union. Its <probability> is at least $1/2$. Reflection symmetry and the <union bound> give a <probability> at least $1/4$ of reaching the upper path $\Gamma$ through $D$. Conditional opening of the <edges> on $\Gamma$ can only help. Since $V(S)$ has <probability> at least $1/2$, the <increasing event> $B$ that a vertical crossing of $S$ is connected inside $T$ to its left side has <probability> at least $1/8$.
Reflect this construction in the vertical midline of $S$ to obtain an <increasing event> $B'$ of the same <probability>, connecting a vertical crossing of $S$ to the right side of $[0,2n]\times[-n,n]$. On $H(S)\cap B\cap B'$, the horizontal crossing of $S$ meets both vertical crossings, so all three <graph paths> join and give $H(R)$. The <Harris-FKG inequality> now yields
$$
\mathbb P_{1/2}(H(R))\geq\frac12\left(\frac18\right)^2=\frac1{128}.
$$
The exploration only conditions the exposed side of the extremal <graph path>; reflecting a drawn path does not assert that its reflected <edges> are open. This distinction is what makes the extension argument valid.
To pass from aspect ratio $3/2$ to $3$, put four rectangles of width $3h/2$ and height $h$ next to each other, with successive left boundaries separated by $h/2$. Adjacent rectangles overlap in a square of side $h$. Require horizontal crossings of the four rectangles and vertical crossings of the three overlap squares. Every consecutive pair of horizontal crossings meets the vertical crossing in its overlap; together they form a horizontal crossing of width $3h$. The <Harris-FKG inequality> gives a strictly positive bound, for these compatible scales, of
$$
c_*=(1/128)^4(1/2)^3.
$$
The construction works on the rotated <square lattice> with the boundary sites used in the source drawing. Integer roundings do not create a scale restriction: for an odd height use the largest smaller compatible even height, and use eight horizontal rectangles and seven overlap squares, whose union has auxiliary aspect ratio $5$, so that its width exceeds the desired $3m$ width for all sufficiently large $m$. Truncate its horizontal crossing on first reaching the desired right side. The number of extra rectangles and overlap-square crossings is bounded independently of $m$, and every factor is bounded below by the same positive constants. The finitely many smallest rectangles each have positive crossing <probability>, since a specified finite open <graph path> suffices. Therefore a constant $c>0$ works at every integer scale. Taking $\tau=c/2$ ensures the requested strict inequality:
$$
\boxed{\inf_{m\geq1}\mathbb P_{1/2}(C_{3m,m})\geq c>\tau>0}.
$$
This proves the required instance of the <Russo-Seymour-Welsh theorem> through reflection, exploration and gluing. The parameter in this calculation is $1/2$.
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