Let $R$ be a noetherian local ring; I say it has isolated singularity if its spectrum is regular outside the closed point. Such rings certainly don't need to be irreducible, for example the localisation of $k[x,y]/(x,y)$ at $(x,y)$. However, in dimension 2 I have to work a little harder to build an example; the first that springs to mind is 'the union of two planes through the origin in $\mathbb A^4$', which I write more formally as the localisation of $$k[w,x,y,z]/((w, x)(y,z))$$ at $(w, x,y,z)$. However, this is a rather horrible ring (in particular not lci). In fact, I've just about convinced myself that this cannot happen for hyper surfaces:
Claim: Let $R$ be a regular local ring of dimension at least 3, and let $r \in R$ be non-zero and such that $R/(r)$ has isolated singularity. Then $R/(r)$ is irreducible.
(Idea of proof: otherwise we factor $r = r_1r_2$, let's imagine distinct irreducibles for simplicity; then the intersection of $V(r_1)$ and $V(r_2)$ should have codimension at most 1 in either of them).
This motivates my question:
Let $R$ be a complete intersection local ring of dimension at least 2 with isolated singularity. Does it follow that $R$ is irreducible?
If I had to guess my money would be somewhat on the 'no' side, but if it were true I could shorten a proof, so seems worth asking. Thanks in advance for any thoughts!
(p.s. by 'irreducible' I mean 'every zero-divisor is nilpotent').