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:

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                    [![Snake][1]][1]

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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.