Subrepresentation conjecture for fine Selmer finiteness

Let $K$ be a number field, let $p$ be a prime, and let $M$ be a representation of $\operatorname{G}_K$ on a finite-dimensional vector space over $\mathbb{F}_p$. For such a representation, let $A(K,M)$ denote the associated fine Selmer finiteness statement. Subrepresentation conjecture. For every subrepresentation $N\subset M$,

$$A(K,M)\ \text{implies}\ A(K,N).$$

The conjecture would give downward inheritance of fine Selmer finiteness; the source discusses several cases and consequences but leaves the general assertion open.