Egan and Nikolayevsky's conjecture on degree sequences of triangular simple graphs

Let $n\geq 3$ and let $(d_1,\ldots,d_n)$ be a sequence of integers satisfying

$$d_1\geq\cdots\geq d_n\geq 4,$$ $$d_1+\cdots+d_n\text{ is even},$$

and

$$\sum_{i=1}^k d_i\leq k(k-1)+\sum_{i=k+1}^n\min\{d_i,k\}\quad\text{for every }k\in[n].$$

Here a triangular simple graph is a simple graph in which every edge is contained in a triangle. Egan and Nikolayevsky's conjecture. The sequence $(d_1,\ldots,d_n)$ is the degree sequence of a triangular simple graph. This is the simple-graph degree-sequence conjecture that motivates the multigraph analogue studied in the paper; the cited authors proved it in some special cases, but the general assertion is not resolved in the supplied text.