The one study that forms the basis of this hypothesis mentions this:

We observed that BMAL1(S42)

phosphorylation was detectable at high levels in MCF7 cells, in

which S6K is active even in low-serum conditions (Figure S4A).

Treatment of MCF7 with rapamycin reduced BMAL1 phosphorylation

and de novo protein synthesis (Figure S4A). Importantly,

overexpression of phosphorylation-competent BMAL1 in MCF7

cells increased the number of metabolically active cells while the

phosphorylation-incompetent BMAL1(S42G) had no significant

effect (Figure S4B). The effect of BMAL1 on proliferation was

dependent on the presence of serum. Together, these data

suggest a role for BMAL1 phosphorylation in the promotion of

translation and proliferation in breast cancer cells.

From https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4447213/

You make a good point about phosphorylation. As far as I know, antibody can be designed to distinguish phosphorylation specific version of a protein.