Hypothesis Testing

Sir,
In this question the solution has been given such that equality of means of independent samples is tested with a t-statistic when Population Variances are unkown but assumed unequal.

But sir, in the core as well as in class, we operate under an implicit assumption thta both these populations have equal variance for the purpose of obtained a pooled estimate of common variance and then proceed to test the hypothesis.

So, sir would you pls clarify why this conflict b/w this answer and Core cum our classes ?