The edge spectral extremal problem for $k{K}_3$ in nonzero residue classes

For a fixed integer $k\ge 2$, let $k{K}_3$ 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 $k{K}_3$-free graph $G$ of size $m$ satisfies $\lambda (G)\le (k-1)+\sqrt{m-k(k-1)},$ and equality holds if and only if $(2k-1)\mid m$ and $G\cong (K_{2k-1}\vee q{K}_1)\cup t{K}_1,q=\frac{m}{2k-1}-(k-1)$ for some $t\ge 0$. They explicitly posed the open problem: Let $k\ge 2$ be fixed and $\mathrm{\ell }$ be a residue in $\{1,\dots ,2k-2\}$. For all sufficiently large integers $m\equiv \mathrm{\ell }(\mathrm{m}\mathrm{o}\mathrm{d}2k-1)$, determine the exact value of $\max \{\lambda (G):e(G)=m,G\text{ is }k{K}_3\text{-free}\},$ and characterize all graphs attaining this maximum. In this paper, using the positive-defect version of the bounded-core method for divisible sizes together with several new ideas developed in this paper, we give a complete solution to the aforementioned open problem.