Eventual multiplicativity of algorithmic complexity under parity-perturbed composition

For Boolean functions $f$ on $n$ bits and $g$ on $m$ bits, let $f\circ g$ denote their block composition, let $\operatorname{PAR}_k$ be parity on $k$ bits, let $g\oplus\operatorname{PAR}_k$ denote the corresponding XOR construction, and let $a(h,\pi_{1/2})$ be the distributional algorithmic complexity of $h$ under the uniform distribution.

Composition multiplicativity conjecture. For every $n,m\ge1$, there exists $k_0(n,m)\in\mathbb N$ such that, for every Boolean functions $f$ and $g$ on $n$ and $m$ bits, respectively, and every $k\ge k_0(n,m)$,

$$a\left(f\circ\left(g\oplus\operatorname{PAR}_k\right),\pi_{1/2}\right)=a(f,\pi_{1/2})\cdot a\left(g\oplus\operatorname{PAR}_k,\pi_{1/2}\right).$$

The conjecture is motivated by the preceding composition theorem and is intended to connect distributional algorithmic complexity with asymptotic separation questions. Its status remains open.