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 $D_4^{(3)}$. Along the same line, we arrive at three partition theorems which may be regarded as an $A_4^{(2)}$ analog of the conjecture. One had been proven by Andrews-van Ekeren-Heluani and we point out that the others are easily proved from their results.