Are there only finitely many unitary perfect numbers? (Erdős #1052)

A unitary divisor of $n$ is a divisor $d\mid n$ with $\gcd(d,n/d)=1$. Call $n\geq 1$ a unitary perfect number if it equals the sum of its unitary divisors other than $n$ itself; equivalently $\sigma^{*}(n)=2n$, where $\sigma^{*}(n)=\sum_{d\mid n,\ \gcd(d,n/d)=1} d$ is the sum-of-unitary-divisors function. Are there only finitely many unitary perfect numbers?

Acceptance. FULLY RESOLVES (OPEN, proof-shaped): a complete proof that only finitely many unitary perfect numbers exist (or a proof that infinitely many do). A single new example does not settle finiteness. ADVANCES: exhibit a sixth unitary perfect number as a program-checkable witness — give its prime factorisation and verify $\sigma^{*}(n)=2n$ (this is a genuine record and collects Subbarao's per-example bounty); OR extend the exhaustive search with reproducible code plus a certificate that no unitary perfect number exists in a stated range beyond the current verified bound; OR a proof, under a clearly stated hypothesis, bounding their number. Deliver the witness with its factorisation and $\sigma^{*}$ check, the search program together with the certified range, or the proof.