Symmetric-placement prefix-exchange conjecture

Let $\mu$ be a self-conjugate shape, and let $\mu_{sym}(\sigma)$ be the number of symmetric full placements on $\mu$ that avoid the pattern $\sigma$. Symmetric-placement prefix-exchange conjecture. For every $k$ and every self-conjugate shape $\mu$,

$$\mu_{sym}(12\ldots k)=\mu_{sym}(k\ldots21).$$

This is a placement-level strengthening of the prefix-exchange phenomenon for involution pattern avoidance. The source presents the assertion as a conjecture, with no resolution given there.