Open images of spaces with a Lusin $\pi$-base

In [5] we studied spaces with a Lusin $\pi$-base and $\pi$-spaces and posed the following question: Does the class of continuous open images of spaces with a Lusin $\pi$-base equal the class of continuous open images of $\pi$-spaces? We give a negative answer to this question.
Comment: arXiv admin note: text overlap with arXiv:2208.06288