Degree conjecture for reduced multiplication and powering polynomials

Let $\hat{F}_n(T;x,y)$ and $\hat{K}_n(T;x,z)$ denote the remainders of the multiplication and powering polynomials modulo the ideal generated by the associativity-defect coefficients. Here degree means the maximal total degree in the indeterminates $x,y,z$, with the parameters $T$ treated as parameters.

Degree conjecture. For every relevant $n$,

$$\hat{F}_n(T;x,y)\text{ has degree }n-1$$

and

$$\hat{K}_n(T;x,z)\text{ has degree }2(n-1).$$

The conjecture is motivated by the computations reported for $1\leq n\leq 7$. The supplied text gives experimental evidence but no proof or resolution.