Brown's converse conjecture on concave renewal functions and DFR distributions

Let $F(t)$ be a distribution function on $\mathbf{R}_+=[0,\infty)$ with $F(0)=0$, and let $M(t)$ be the renewal function satisfying

$$M(t)=F(t)+\int_0^t M(t-x)\,\mathrm{d}F(x),\quad t\geq 0.$$

A distribution on $\mathbf{R}_+$ is DFR (decreasing failure rate) when its survival function is log-convex on $\mathbf{R}_+$. Brown's conjecture. If the renewal function $M(t)$ is concave on $\mathbf{R}_+$, then $F(t)$ is DFR. The forward implication, that DFR implies concavity of the renewal function, is known; Brown's converse question asks whether concavity is sufficient for the DFR property.