Order drop, modular identification, and a conjectural supercongruence for Sym³ hypergeometric coefficients

The Mao–Tian order-3 recurrence for coefficients of ₂F₁(a,b;c;z)³ drops to order 2 at (1/3,1/3,1). The rescaled sequence A_n = 27^n [z^n] ₂F₁(1/3,1/3;1;z)³ equals, after sign twist, the eta-quotient η(τ)⁹/η(3τ)³ in the Hauptmodul of X₀(3). A conjectural supercongruence A(mp) ≡ A(m) mod p⁴ is reduced to a Dwork-type input and verified for p ≤ 47. The exponent 4 is sharp.