We can study the conjecture for locally compact rings as follows. 

In [Kap] a subset $S$ of a topological ring is called *algebraically nilpotent* if for some $n$, $S^n=0$.

**Lemma 1.** [Kap, Theorem 2] A  locally  compact  ring with  no algebraically  nilpotent ideals  is the  direct  sum  of  a connected  ring  and a totally  disconnected  ring. The former  is a semi-simple  algebra  of  finite  order  over  the  real  numbers.

Since Romain Gicquaud's [comment](https://mathoverflow.net/questions/454884/are-topological-pids-noetherian#comment1177754_454894) rules out the connected ring case, it remains to consider locally compact totally disconnected rings. For this case we have the following structural lemma.

**Lemma 2.** [Kap, Lemma 4]  A locally  compact  totally  disconnected  ring  has  a system of  neighborhoods  of  $0$ which  are compact open subrings.

**Lemma 3.** Let $R$ be a locally  compact  totally  disconnected integral domain. Let $U$ be a compact open subring of $R$ such that $0\in U$ and $1\not\in U$. Then $1+U$ is a multiplicative topological group.

*Proof.* Since $U$ is a ring, $1+U$ is a multiplicative semigroup. Since $R$ is an integral domain and $0\not\in 1+U$, the multiplicative semigroup $1+U$ is cancellative. Since $1+U$ is compact, it is a multiplicative topological group, see, for instance, [AT, Theorem 2.5.2]. $\square$

**Proposition 4.** Let $R$ be a locally compact totally disconnected integral domain which is a union of less than $\mathfrak c$ many compact sets. If every closed ideal in $R$ is principal then each proper nonzero ideal $I$ of $R$ is closed, so $R$ is a principal ideal domain.

*Proof*. Pick an element $x\in R$ such that $(x)=\overline{I}$. By Lemma 2, there exists a compact open subring $U$ of $R$ such that $0\in U$ and $1\not\in U$. There exists a set $A$ of size less than $\mathfrak c$ and the family $\{K_\alpha:\alpha\in A\}$ of compact subsets of $R$ such that $R=\bigcup_{\alpha\in A} K_\alpha$. For each $\alpha\in A$ the family $\{a+U:a\in K_\alpha\}$ is an open cover of a compact set $K_\alpha$. Therefore there exists a finite subset $S_\alpha$ of $K_\alpha$ such that $K_\alpha\subset S_\alpha+U$. Put $S=\bigcup_{\alpha\in A} S_\alpha$. Then $|S|\le |A|\cdot \omega<\mathfrak c$ and $S+U=R$. Let $T$, $T=xR/xU$ be the quotient additive topological group and $q:xR\to T$ be the quotient map. By [AT, Proposition 3.1.23] the topological group $T$ is locally compact. Moreover, $T=xR/xU=q(xR)=q(x(S+U))=q(xS+xU)=q(xS)$, so $|T|<\mathfrak c$. Since a Hausdorff compact space without isolated points has size at least $\mathfrak c$, for instance, by Čech–Pospíšil theorem, see, for instance, [Hod, Theorem 7.19], the group $T$ is discrete, so the set $xU=q^{-1}(0_T)$ is a neighborhood of $0$ in the additive topological group $xR$. Therefore $x+xU$ is a neighborhood of $x$ in $xR$, so there exists a point $y\in I\cap (x+xU)$. That is $y=x+xu$ for some element $u\in U$. By Lemma 3, $1+U$ is a multiplicative topological group, so the element $1+u$ is invertible. Then $x\in I$, so $I=(x)=\overline{I}$. $\square$

Now we can describe the structure of compact topological principal ideal domains in more details applying the following results. 

**Proposition 5.** [War, Theorem 1] Let $R$ be a compact  Hausdorff ring, and let $\mathfrak r$ be the radical of $R$. Every ideal of $R$  is  closed if and  only  if $R$  satisfies  the ascending  chain  condition  on  ideals and  every  principal  ideal  of $R$ is closed.  Under these circumstances,  the  topology of $R$  is  the  $\mathfrak r$-adic  topology,  either  $R=\mathfrak r$  or $R/\mathfrak r$  is  isomorphic  to  the  Cartesian product of a  finite family  of finite  simple  rings,  and  consequently  there are  only finitely  many  regular  maximal,  regular  maximal  left,  or  regular  maximal  right ideals in $A$.

**Proposition 6.** [War, Theorem 2] Let $R$ be a  topological  ring with  identity,  and  let $\mathfrak r$  be  the  radical of $R$. The following  conditions  are equivalent.

1)) $R$  is  compact Hausdorff,  and  every  left  ideal  of $R$  is  closed.

2)) $R$  is  compact Hausdorff and  Noetherian.

3))  $R$  is compact Hausdorff and satisfies the ascending chain condition on closed  left ideals.

4))  $R$  is  Noetherian,  the  topology  of $R$  is  the  $\mathfrak r$-adic  topology,  $R$  is  complete for  that  topology, $\bigcap_{n=1}^\infty \mathfrak r^n=\{0\}$, and $R/\mathfrak r$  is a  finite  ring.


*References*

[AT] Alexander V. Arhangel'skii, Mikhail G. Tkachenko, *Topological groups and related structures*, Atlantis Press, Paris; World Sci. Publ., NJ, 2008.

[Eng]  Ryszard Engelking, *General Topology*, 2nd ed., Heldermann, Berlin, 1989.

[Hod] R. Hodel, *Cardinal Functions I*, in: K.Kunen, J.E.Vaughan (eds.) *Handbook of Set-theoretic Topology*, Elsevier Science Publishers B.V., 1984.

[Kap] Irving Kaplansky, *[Locally compact rings](https://www.jstor.org/stable/2372343)*, Am. J. Math. **70**:2 (Apr 1948) 447-459.

[War] Seth Warner, *[Compact rings](https://link.springer.com/article/10.1007/BF01452361)*, Mathematische Annalen **145** (Feb 1962) 52-63.