Let $(p_1, p_2)$ be a twin prime pair, where we include $(2, 3)$. If $p_1 \equiv 1$ mod $4$ then we let $t_{(p_1, p_2)} := p_1 ^ 2 / p_2 ^ 2$ otherwise, we let $t_{(p_1, p_2)} := p_2 ^ 2 / p_1 ^ 2$.

I conjecture that the product $$ \prod_{(p_1, p_2): \text{twin primes}}t_{(p_1, p_2)} =\tfrac{3 ^ 2}{2 ^ 2} \cdot \tfrac{5 ^ 2}{3 ^ 2}\cdot \tfrac{5 ^ 2}{7 ^ 2}\cdot\tfrac{13 ^ 2}{11 ^ 2} \cdot\tfrac{17 ^ 2}{19 ^ 2} \cdot\tfrac{29 ^ 2}{31 ^ 2} \cdot\tfrac{41 ^ 2}{43 ^ 2} \cdot \tfrac{61 ^ 2 }{59 ^ 2} \cdot \tfrac{73 ^ 2}{ 71 ^ 2}\cdot \tfrac{101 ^ 2 }{ 103 ^ 2}\cdots $$

is equal to $\pi$. (If this is true then twin prime numbers are infinity many.)

Some numerical values of partial products:

___________________________ $p_1 \equiv 3$ mod $4$ ___ $p_1 \equiv 1$ mod $4$

3.1887755102040816321 to $10^1$, ___________ 1 = ____________ 1

3.2055606708805624550 to $10^2$, ___________ 4 = ____________ 4

3.1290622219773513145 to $10^3$, __________ 16 < ___________ 19

3.1364540609918890779 to $10^4$, _________ 100 < __________ 105

3.1384537326021492746 to $10^5$, _________ 620 > __________ 604

3.1417076006640026373 to $10^6$, ________ 4123 > _________ 4046

3.1417823471756806475 to $10^7$, _______ 29498 > ________ 29482

3.1415377533170544536 to $10^8$, ______ 219893 < _______ 220419

3.1415215264211035597 to $10^9$, _____ 1711775 < ______ 1712731

3.1415248453830039795 to $10^{10}$, ____13706087 < _____ 13706592

3.1415126339547108140 to $10^{11}$,

3.1415144504088659201 to $10^{12}$,

3.1415142045284687040 to $10^{13}$,

3.1415144719058962626 to $10^{14}$,

3.1415384423175311229 to $10^{15}$

Can we find a few more decimal places using the extrapolation method?