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This looks like the kind of thing that one might be able to prove by filtering $U$ by word-length over $g$ and then passing to the associated graded ring. The idea is that this would reduce the problem to the case that $g$ is abelian, since the associated graded ring is a polynomial ring in $\dim g$ variables and the number of generators of $I$ as a left ideal will be at least as big as the number of generators of $\mathrm{gr}(I)$ its associated graded ideal.

I'm not sure whether the invariant dimension will be preserved when you pass to the associated graded though. You might expect it to be as a similar invariant, the grade of the module M, j(M) is preserved when you do this. (Recall that j(M) is the smallest integer j such that $Ext^j_U(M,U)\neq 0$)

Edit: Expansion of strategy:

Proposition 7.1 of Auslander-Gorenstein Rings by Ajitabh, Smith and Zhang tells us that $\mathrm{Ext}^j_U(M,k)\cong \mathrm{Ext}^{d-j}_U(k,M)^*$ where $d=\dim g$ and $(-)^*$ means $k$-vector space dual. Thus invariant dimension is $d-min(j|\mathrm{Ext}^j_U(M,k)\neq 0)$. If one can relate the non-vanishing of $\mathrm{Ext}^j_U(M,U)$ and $\mathrm{Ext}^j(M,k)$ then it might be possible to reduce the problem to something relating the grade of $M$ to the number of relations of $M$. This can be done by passing to the associated graded ring.

This looks like the kind of thing that one might be able to prove by filtering $U$ by word-length over $g$ and then passing to the associated graded ring. The idea is that this would reduce the problem to the case that $g$ is abelian, since the associated graded ring is a polynomial ring in $\dim g$ variables and the number of generators of $I$ as a left ideal will be at least as big as the number of generators of $\mathrm{gr}(I)$ its associated graded ideal.

I'm not sure whether the invariant dimension will be preserved when you pass to the associated graded though. You might expect it to be as a similar invariant, the grade of the module M, j(M) is preserved when you do this. (Recall that j(M) is the smallest integer j such that $Ext^j_U(M,U)\neq 0$)

Edit: Expansion of strategy:

Proposition 7.1 of Auslander-Gorenstein Rings by Ajitabh, Smith and Zhang tells us that $\mathrm{Ext}^j_U(M,k)$ is isomorphic to the $k$-vectorspace dual of $\mathrm{Ext}^{d-j}_U(k,M)$ where $d=\dim g$. Thus invariant dimension is $d-min(j|\mathrm{Ext}^j_U(M,k)\neq 0)$. If one can relate the non-vanishing of $\mathrm{Ext}^j_U(M,U)$ and $\mathrm{Ext}^j(M,k)$ then it might be possible to reduce the problem to something relating the grade of $M$ to the number of relations of $M$. This can be done by passing to the associated graded ring.

This looks like the kind of thing that one might be able to prove by filtering U by word-length over g and then passing to the associated graded ring. The idea is that this would reduce the problem to the case that g is abelian, since the associated graded ring is a polynomial ring in $\dim$ g variables and the number of generators of I as a left ideal will be at least as big as the number of generators of gr(I) its associated graded ideal.

I'm not sure whether the invariant dimension will be preserved when you pass to the associated graded though. You might expect it to be as --- if I remember correctly --- a similar invariant, the grade of the cyclic module M, j(M) is preserved when you do this. (Recall that j(M) is the smallest integer j such that $Ext^j_U(M,U)\neq 0$)

This looks like the kind of thing that one might be able to prove by filtering $U$ by word-length over $g$ and then passing to the associated graded ring. The idea is that this would reduce the problem to the case that $g$ is abelian, since the associated graded ring is a polynomial ring in $\dim g$ variables and the number of generators of $I$ as a left ideal will be at least as big as the number of generators of $\mathrm{gr}(I)$ its associated graded ideal.

I'm not sure whether the invariant dimension will be preserved when you pass to the associated graded though. You might expect it to be as a similar invariant, the grade of the module M, j(M) is preserved when you do this. (Recall that j(M) is the smallest integer j such that $Ext^j_U(M,U)\neq 0$)

Edit: Expansion of strategy:

Proposition 7.1 of Auslander-Gorenstein Rings by Ajitabh, Smith and Zhang tells us that $\mathrm{Ext}^j_U(M,k)\cong \mathrm{Ext}^{d-j}_U(k,M)^*$ where $d=\dim g$ and $(-)^*$ means $k$-vector space dual. Thus invariant dimension is $d-min(j|\mathrm{Ext}^j_U(M,k)\neq 0)$. If one can relate the non-vanishing of $\mathrm{Ext}^j_U(M,U)$ and $\mathrm{Ext}^j(M,k)$ then it might be possible to reduce the problem to something relating the grade of $M$ to the number of relations of $M$. This can be done by passing to the associated graded ring.

This looks like the kind of this that one might be able to prove by filtering U by word-length over g and then passing to the associated graded ring. The idea is that this would reduce the problem to the case that g is abelian, since the associated graded ring is a polynomial ring in $\dim$ g variables and the number of generators of I as a left ideal will be at least as big as the number of generators of gr(I) its associated graded ideal.

I'm not sure whether the invariant dimension will be preserved when you pass to the associated graded though. You might expect it to be as --- if I remember correctly --- a similar invariant, the grade of the cyclic module M, j(M) is preserved when you do this. (Recall that j(M) is the smallest integer j such that $Ext^j_U(M,U)\neq 0$)

This looks like the kind of thing that one might be able to prove by filtering U by word-length over g and then passing to the associated graded ring. The idea is that this would reduce the problem to the case that g is abelian, since the associated graded ring is a polynomial ring in $\dim$ g variables and the number of generators of I as a left ideal will be at least as big as the number of generators of gr(I) its associated graded ideal.

I'm not sure whether the invariant dimension will be preserved when you pass to the associated graded though. You might expect it to be as --- if I remember correctly --- a similar invariant, the grade of the cyclic module M, j(M) is preserved when you do this. (Recall that j(M) is the smallest integer j such that $Ext^j_U(M,U)\neq 0$)