The edge spectral extremal problem for $kK_3$ in nonzero residue classes

For a fixed integer $k\ge 2$, let $kK_3$ denote the vertex-disjoint union of $k$ triangles. A recent fixed-size spectral theorem of Das and Yamini asserts that, for all sufficiently large $m$, every $kK_3$-free graph $G$ of size $m$ satisfies $λ(G)\le (k-1)+\sqrt{m-k(k-1)},$ and equality holds if an