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We prove a strong factorization property of interpolation Macdonaldpolynomials when $q$ tends to $1$. As a consequence, we show that Macdonaldpolynomials have a strong factorization property when $q$ tends to $1$, whichwas posed as an open question in our previous paper with F\'eray. Furthermore,we introduce multivariate $q,t$-Kostka numbers and we show that they arepolynomials in $q,t$ with integer coefficients by using the strongfactorization property of Macdonald polynomials. We conjecture thatmultivariate $q,t$-Kostka numbers are in fact polynomials in $q,t$ withnonnegative integer coefficients, which generalizes the celebrated Macdonald'spositivity conjecture.

Strong factorization property of Macdonald polynomials and higher-order Macdonald's positivity conjecture