#### Does this infinite primes snake-product converge?

This re-asks a question I posed on MSE:

Q. Does this infinite product converge?

$$\frac{2}{3}\cdot\frac{7}{5}\cdot\frac{11}{13}\cdot\frac{19}{17}\cdot\frac{23}{29}\cdot\frac{37}{31} \cdot \cdots \;.$$ I call this the primes snake-product:

Out to primes of size $$10^{10}$$, MSE user @Peter calculated the product to be $$\approx 0.9048$$.

@Wojowu showed that the question is likely difficult, relying on estimates of alternating sums of prime gaps, and that perhaps convergence is beyond current knowledge. I re-pose the question to see if indeed this is the case, or might known bounds suffice to establish convergence.