A vertex operator reformulation of the Kanade–Russell conjecture modulo 9

We reformulate the Kanade–Russell conjecture modulo 9 via the vertex operators for the level 3 standard modules of type \(D^{(3)}_{4}\) $D_4^{(3)}$ . Along the same lines, we arrive at three partition theorems which may be regarded as an \(A^{(2)}_{4}\) $A_4^{(2)}$ analog of the conjecture. Andrews–van Ekeren–Heluani have proven one of them, and we point out that the others are easily proven from their results.