Let $A$ be an abelian variety over a field $k$ of dimension $g$, and $H$ be a Weil cohomology theory for smooth projective varieties over $k$ with characteristic $0$ coefficient field $E$.

Is it true that $\operatorname{dim} H^1(A)=2g$? I think this will follow from some standard conjectures, but do we know this unconditionally at present (at least for some low dimensional cases)?

For example, one can prove $\operatorname{dim} H^1(A) \leq 2g$ by some power tricks, see Chapter 3. Theorem 8.1 in https://pages.uoregon.edu/ddugger/wbook.pdf (it may contain some typos, but the idea works).

And if $k$ is algebraically closed and $A$ is superspecial, then $End_k(A)\otimes \mathbb Q \cong M_g(D_{p,\infty})$ is central simple algebra of dimension $4g^2$ over $\mathbb Q$, hence $End(A) \otimes E \hookrightarrow End(H^1(A))$, and $dim H^1(A)=2g$.