The connected-heavy-edge conjecture for $T_5$-factors

Let $M$ be a standard multigraph on $n$ vertices, where $n$ is divisible by $3$. Let $\delta(M)$ denote its minimum degree, let $H_M$ be its heavy-edge graph, and let a $T_5$-factor mean a factor consisting of copies of $T_5$. The connected-heavy-edge conjecture asserts that if

$$\delta(M)\ge \frac{4}{3}n-1$$

and $H_M$ is connected, then $M$ has a $T_5$-factor. The conjecture is presented as a possible strengthening suggested by an extremal example; the supplied text does not state that it has been proved or disproved.