Closure of the set of Markov stopping-time measures under convolution and convex combination

Let $\mathfrak{P}$ denote the set of measures considered in the paper that arise directly or indirectly, after extension, from a Markov stopping time. Closure conjecture. The set $\mathfrak{P}$ is closed under convolution and convex combination. The paper establishes closure under infinite convex combination and convolution for measures arising directly or indirectly from a Markov stopping time, but does not establish these closure properties for all of $\mathfrak{P}$; their validity remains open.