Dwork-type supercongruences for the truncated series 2F1(r/d, 1−r/d; 1; 1) via a multiplicative q-congruence

For integers d ≥ 2 and 1 ≤ r ≤ d−1 let S_N(r) = Σk<N (r/d)_k (1−r/d)_k / k!² be the truncation of the hypergeometric series 2F1(r/d, 1−r/d; 1; 1) after N terms. For r/d = 1/2, 1/3, 1/4, 1/6 these are the truncated sums of C(2k,k)²/16^k, C(2k,k)C(3k,k)/27^k, C(2k,k)C(4k,2k)/64^k and C(3k,k)C(6k,3k)/432^k, whose supercongruences were observed by Rodriguez-Villegas and studied by Mortenson, Z.-H. Sun, J.-C. Liu, Guo, Zudilin and others. We prove the Dwork-type supercongruences Smp^s(r) ≡ (−1)^⟨−r/d⟩_p · S_{mpˢ⁻¹}(r_p) (mod p²ˢ) for every odd prime p not dividing d and all integers m, s ≥ 1, where r_p/d is the Dwork prime of r/d, that is, r_p ≡ r/p (mod d). Equivalently, Sₚₙ(r) − (−1)^⟨−r/d⟩_p · S_n(r_p) is divisible by p2+2v_p(n) for every n ≥ 1, and the exponent is best possible. For the four values of Rodriguez-Villegas the two sums coincide, and we obtain the towers Smp^s ≡ χ(p) S_{mpˢ⁻¹} (mod p²ˢ) with χ(p) = (−1/p), (−3/p), (−2/p), (−1/p) respectively. This extends results of Mortenson and Z.-H. Sun (m = s = 1), J.-C. Liu (s = 1, modulo p²) and H.-X. Ni (m = 1, p ≡ ±1 mod d). It proves the integrality part of a conjecture of Z.-W. Sun (Conjecture 5.4 in arXiv:1610.03384) for every n for which the binomial factor in Sun's normalization is prime to p, and it explains Liu's observation that his congruence does not extend to arbitrary parameters: the companion of Sₚₙ(r) is S_n(r_p), with the Dwork prime. The proof goes through a q-analogue. For K_N(r;q) = Σ_{k