The Lie-algebra cohomology conjecture for nilpotent subalgebras

Let $K$ be a nonarchimedean field of mixed characteristics $(0,p)$. Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over $K$, let $\mathfrak{h}$ be a nilpotent Lie subalgebra of $\mathfrak{g}$, and let $M$ be a Banach space over $K$ equipped with a $K$-linear action of $\mathfrak{g}$.

Lie-algebra cohomology conjecture. (a) If

$$H^i(\mathfrak{h},M)=0$$

for all $i\geq 0, then

$$H^i(\mathfrak{g},M)=0$$

for all $i\geq 0$. (b) If

$$\dim_K H^i(\mathfrak{h},M)<\infty$$

for all $i\geq 0, then

$$\dim_K H^i(\mathfrak{g},M)<\infty$$

for all $i\geq 0$.

This conjecture is the Lie-algebra analogue proposed as a reduction of the mixed-characteristic analytic conjecture. The Hochschild–Serre spectral sequence gives the corresponding implication for subnormal inclusions, but the stated nilpotent-subalgebra claim remains open.