Kaplansky (1951)'s original definition of an AW$^*$-algebra is a C$^*$-algebra $A$ such that (i) each family of orthogonal projections has a least upper bound (LUB) in $P(A)$ (the set of projections in $A$) and (ii) each maximal abelian $*$-subalgebra (masa) of $A$ is generated by its projections.
Moreover, a C$^*$-algebra $A$ is said to be monotone complete if every increasing, norm-bounded net in $A_{sa}$ has a LUB in $A_{sa}$. Here $A_{sa}$ refers to the algebra of self-adjoint elements of $A$.
In 2015, Saito and Wright, published a proof of a longstanding folklore result that if $A$ is a C$^*$-algebra in which every masa is monotone complete, then $A$ is a (unital) AW$^*$-algebra. They note that this is, in fact, an "if and only if" statement, saying:
When A is an AW$^*$-algebra it can be proved that each maximal abelian *-subalgebra of A is monotone complete and A is unital.
There is no reference provided. Is there a way to show this direction of the statement by appealing directly to Kaplansky's original definition?
With considerable effort, one can show that for a commutative C$^*$-algebra $A$, TFAE:
(i) $A$ is an AW$^*$-algebra, (ii) $A$ is monotone complete, and (iii) $A \cong C(X)$ where $X$ is a compact, Hausdorff, extremally disconnected space.
I have (only) seen proofs of (i) $\iff$ (iii) and (ii) $\iff$ (iii). A direct proof of (i) $\implies$ (ii), which appeals to the original definition, would prove the claim, because every masa of an AW$^{*}$-algebra is an AW$^{*}$-algebra (which Kaplansky showed in his original paper).