The maximal-dimension conjecture for 1-spec matrix spaces

Let $V$ be a $1$-spec subspace of $\operatorname{M}_n(\mathbb{K})$ such that

$$\dim V=1+\binom{n}{2}.$$

Assume $n>4$. A linear subspace $W$ of $\operatorname{M}_n(\mathbb{K})$ has a trivial spectrum if no element of $W$ has a nonzero eigenvalue. Maximal-dimension conjecture. There exists a linear subspace $W$ of $\operatorname{M}_n(\mathbb{K})$ with a trivial spectrum such that

$$V=\mathbb{K}I_n+W.$$

The conjecture concerns the classification of $1$-spec subspaces of maximal dimension and, if true, would resolve the question for $\#\mathbb{K}>2$ and $n>4$ using the cited results on affine spaces of nonsingular matrices. Its status is not established in the supplied text.