When in doubt about your assumption that the population variances are equal when comparing two population means, the best strategy is to proceed with the unequal variances test?

That statement is true, the best strategy to do next is indeed the unequal variance test.
The unequal variances test is only used when the two populations that were tested are equal.
This test that will crossreference every variances that these populations have in order to spot the difference that may cause the difference within the data