Levchuk's Problem on Diagonal Matrices in SL_n(q) (Kourovka Notebook 7.27): An Affirmative Answer as a Corollary of Theorems of Fulman and Guralnick

V. M. Levchuk asked in 1980 (Kourovka Notebook, Problem 7.27) whether, for all sufficiently large \(q\), the group \(\mathrm{SL}_n(q)\) contains a diagonal matrix lying in no proper irreducible subgroup except block-monomial ones. We show that the answer is yes for every \(n\), as a short corollary of density theorems of Fulman and Guralnick; it is not presented as a new theorem. For fixed \(n\) and \(q\to\infty\), almost all regular diagonal matrices \(d\) of \(\mathrm{SL}_n(q)\) have the property that every proper subgroup containing \(d\) that is irreducible over \(\mathbb{F}_q\) permutes the summands of a decomposition of \(\mathbb{F}_q^n\) into coordinate subspaces. The deduction combines the density theorems with an elementary count of the classes of regular diagonal matrices, a description of the maximal subgroups of maximal rank, and an eigenvalue argument excluding extension field subgroups; the bound for \(q\) is not explicit. For \(n=2\), Dickson's list of subgroups gives the complete answer: a diagonal matrix with this property exists if and only if \(q\in\{4,8\}\) or \(q\ge13\) (if irreducibility is meant absolutely, also for \(q=2\)), and \(\mathrm{diag}(w,w^{-1})\) works for every generator \(w\) of \(\mathbb{F}_q^*\) when \(q\ge13\). No printed statement of the general answer was found, and no priority is claimed. This is an unrefereed note.