Do some of the local cohomology groups of the structure sheaf on the singular locus vanish?

Ok, so this is where I reveal my ignorance as an algebraic geometer: I had previously asked about pushforward of line bundles from the smooth locus of a variety to the whole thing. I think I understand basically how that picture works now, but I have a variation thereof that I would like to ask about.

Let $X$ is a normal variety (let's say irreducible quasi-projective of finite type over $\mathbb{C}$; you can even assume terminal and $\mathbb{Q}$-factorial if you like), $Y$ its smooth locus and $Z$ its singular locus. In fact, let's say the dimension of $Z$ is at most $\dim X - 4$ just for good measure.

What can be said about the local cohomology $H^i_Z(\mathcal{O}_X)$ which fills in the exact sequence $$\cdots \to H^i_Z(\mathcal{O}_X)\to H^i(X;\mathcal{O}_X)\to H^i(Y;\mathcal{O}_Y)\to \cdots?$$

What I'd love to say is that this is 0 in degrees $\leq 3$, but obviously, I'd accept other answers if they happen to be true.

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Basically the point is that what you want is true provided your variety is $S_4$. (See the above link for a proof). In particular, if your variety is Cohen-Macaulay (which terminal and even klt singularities are, but not log canonical) then you're in business.
In general, this vanishing is essentially equivalent to that your variety be $S_4$. If you consider a $4$-dimensional variety $X$ with a single singular point $z\in X$ that is $S_2$, but not $S_4$, then $X$ is normal but $H^i_z(X,\mathscr O_X)\neq 0$.
For an explicit example of such a singularity consider a cone over an abelian threefold. This is a non-klt log canonical singularity. The fact that this is such an example follows from the condition that tells you what $S_m$ a cone is based on the cohomology groups of the scheme it is a cone over. For a complete proof of that criterion in a rather general case see Lemma 3.1 of this paper of Patakfalvi.