A single fixed wire permutation preserves strong simulability

Let $F$ assign to every finite Boolean string $s$ a width $w(s)\in\mathbb N$ and a finite ordered list of gates, each specified by two unitary complex $2\times2$ matrices $A_g,B_g$, indexed by Boolean values, and an injection $v_g:\{0,1\}\hookrightarrow\{0,\ldots,w(s)-1\}$. Assume every gate satisfies $\det A_g=\det B_g$ and $|v_g(0)-v_g(1)|\le1$. On its two designated wires, its matrix entry from $y$ to $x$ is zero when the two strings have different Boolean parity, is $(A_g)_{x_0,y_0}$ when both have even parity, and is $(B_g)_{x_0,y_0}$ when both have odd parity; on all other wires it leaves the bits unchanged. Let $U_s$ be the product of these embedded matrices in reverse list order, with the empty product equal to the identity. For each $s$ and wire $i$, let $\phi_{s,i}\in\mathbb C^2$ have Euclidean norm one, and put $\psi_s(x)=\prod_i\phi_{s,i}(x_i)$. The effectiveness assumptions are as follows. Write $u(j)=\mathrm{true}^{\,j}$ and $T([a_1,\ldots,a_k])=\big\Vert_j(u(|a_j|)\,\mathrm{false}\,a_j)$. The function encoding $F$ as $T([u(w(s)),u(\text{number of gates})]\mathbin{+\!\!+}[T([u(v_g(0)),u(v_g(1))]):g\text{ in list order}])$ is polynomial-time computable. The lists of matrix entries, taking $A_g$ then $B_g$ for each gate and the order $(0,0),(0,1),(1,0),(1,1)$ within each matrix, and the lists $[\phi_{s,0}(0),\phi_{s,0}(1),\ldots]$, each admit polynomial-time numerical approximation: for every valid list index $j$, precision $m\in\mathbb N$, and Boolean $b$, a polynomial-time Boolean-string function queried on $T([s,u(j),u(m),[b]])$ returns a signed dyadic approximation to the real part when $b=\mathrm{false}$, or the imaginary part when $b=\mathrm{true}$, with absolute error at most $2^{-m}$. Here a returned string represents its binary tail, with true as one, divided by $2^m$, negated exactly when its first bit is true; missing bits default to false. All polynomial-time requirements use TM2 machines with identity input and output encodings. Finally, let $p_s$ be a permutation of the wires such that $s\mapsto T([u(p_s(0)),\ldots,u(p_s(w(s)-1))])$ is polynomial-time computable, and define $P_s(x,y)=1$ if $x=y\circ p_s$, and zero otherwise. Then, writing $V_s=P_s^\dagger U_sP_s$, there exists a polynomial-time Boolean-string function which, on $T([s,u(m),u(k),[b]])$, returns a dyadic number within $2^{-m}$ of

for every $s$, $m\in\mathbb N$, wire $k<w(s)$, and Boolean $b$, using the same signed-dyadic convention; moreover, for every $s$, $P_sV_sP_s^\dagger=U_s$. Width zero is included: its product input has its sole amplitude equal to one, and there are no wire-probability queries.