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Motivation: As is stated in the former question, those identities(formula (35)-(44)) of $1/\pi$ attributed to Ramanujan are related to surfaces with Picard rank 20(see the paper of Elkies and Schuett) in the Dwork family $$x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4.$$ Jesus Guillera found a few Ramanujan-type formulas for $1/\pi^2$(which can be found in W. Zudilin's paper), three of which(formula (92)(93)(94) in Zudilin's paper) are related to the Dwork family $$x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6=6\lambda x_1x_2x_3x_4x_5x_6$$through Picard-Fuchs equation. It is reasonable to conjecture that the Guillera's formulas are related to "singular" members in Dwork family.

Motivation: As is stated in the former question, those identities(formula (35)-(44)) of $1/\pi$ attributed to Ramanujan are related to surfaces with Picard rank 20(see the paper of Elkies and Schuett) in the Dwork family $$x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4.$$ Jesus Guillera found a few Ramanujan-type formulas for $1/\pi^2$(which can be found in W. Zudilin's paper), three of which(formula (92)(93)(94) in Zudilin's paper) are related to the Dwork family $$x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6=6\lambda x_1x_2x_3x_4x_5x_6$$through Picard-Fuchs equation. It is reasonable to conjecture that the Guillera's formulas are related to "singular" members in Dwork family.

Update3: It is really a surprise that the formulas are finally proven after 7 years!! See this paper of Kam Cheong Au.

Update2: A recent preprint of David P. Roberts and Fernando Rodriguez Villegas is provided here. It seems that the splitting of underlying hypergeometric motives does not only provide period relations of (generalized) hypergeometric functions at algebraic arguments, but also supercongruences of truncated hypergeometric series.