Yes, it literally states "effectively has", and later states: "Indeed, because of disassortative mating based on both chromosomes 2 and 2m and the W and Z sex chromosomes, the species operates as though there are four sexes."
Only two sex chromosomes, and acts "as though" there are four sexes, which means there aren't four sexes.
The paper does not make the claim that there are four independent sexes. Helmuth incorrectly reported a claim that the paper does not make.
"Squirrels fly through the air as though they are birds" Squirrels are not birds and the previous statement does not support that claim.