The general formula for pattern-avoiding stabilized-interval-free permutations

Let $f_{n,d}^{(2)}(123)$ denote the number of permutations of size $n$ that avoid the pattern $123$ and have stabilized interval parameter $d$. Write $n=2m+r$ with $r\in\{0,1\}$ and let $1\le d\le m$. The general enumeration formula.

$$f_{n,d}^{(2)}(123) = \frac{\binom{n-d}{m}^2}{(n-d)^2} \Bigg[\frac{\binom{2d}{d}d^3(1-r)}{4d-2} + 2\!\sum_{i=1}^{\floor{\frac{m+1}{3}}}\frac{\binom{m}{i-r}\binom{m-d+r}{i}\binom{2d-2}{2i-1+d-r}}{\binom{m+i}{m}\binom{m-d+i}{m-d+r}}(d^2-(2i-r)^2)\Bigg].$$

This formula gives a closed expression for the enumeration in all allowed cases, extending the explicitly computed extreme cases $d=1$ and $d=\floor{n/2}$. The parser provides no evidence resolving whether the asserted formula is proved or remains conjectural.