Conjecture on near-perfect triangle packings in randomly perturbed regular graphs

Let $d>1/2$, let $G_d$ be a $dn$-regular graph on $n$ vertices, and let $G(n,p)$ be the binomial random graph. A triangle packing is a collection of edge-disjoint triangles; it covers all but $o(n^2)$ edges if the number of uncovered edges is $o(n^2)$ as $n\rightarrow\infty$.

Near-perfect packing conjecture. If

$$p>(3-4d)/(4-4d),$$

then with high probability there exists a triangle packing in $G_d\cup G(n,p)$ that covers all but $o(n^2)$ edges.

This conjecture is proposed as the tightness of the paper's lower bound for $p_d$ in the regime $d>1/2$, without assuming the Nash-Williams conjecture. Its resolution is not supplied, so it remains open.