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(I'll just note beforehand that is this is a "I don't know the answer to your actual question, perhaps you'd rather know the answer to a slightly different one" answer). The answer is yes if you replace virtual Betti number with Frobenius trace; these are also motivic, if admittedly slightly harder to calculate, as they involve counting points. This is covered, for example, in Section 3.4 of my second papermy second paper with Geordie Williamson.

(I'll just note beforehand that is this is a "I don't know the answer to your actual question, perhaps you'd rather know the answer to a slightly different one" answer). The answer is yes if you replace virtual Betti number with Frobenius trace; these are also motivic, if admittedly slightly harder to calculate, as they involve counting points. This is covered, for example, in Section 3.4 of my second paper with Geordie Williamson.

(I'll just note beforehand that is this is a "I don't know the answer to your actual question, perhaps you'd rather know the answer to a slightly different one" answer). The answer is yes if you replace virtual Betti number with Frobenius trace; these are also motivic, if admittedly slightly harder to calculate, as they involve counting points. This is covered, for example, in Section 3.4 of my second paper with Geordie Williamson.

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Ben Webster ♦
Ben Webster ♦
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(I'll just note beforehand that is this is a "I don't know the answer to your actual question, perhaps you'd rather know the answer to a slightly different one" answer). The answer is yes if you replace virtual Betti number with Frobenius trace; these are also motivic, if admittedly slightly harder to calculate, as they involve counting points. This is covered, for example, in Section 3.4 of my second paper with Geordie Williamson.