The image/reaction scheme is not provided in the prompt, so the exact structure of compound $X$ cannot be independently derived from the reaction sequence.

However, the stored correct answer is given as $8$. In many standard JEE organic chemistry problems from the chapter alcohols, phenols and ethers, a final product having 8 sp$^2$ hybridised carbons commonly corresponds to a structure containing:

• one benzene ring: $6$ sp$^2$ carbons, and
• one additional double bond or carbonyl-containing unsaturated fragment contributing $2$ more sp$^2$ carbons.

So the total becomes $6 + 2 = 8.$

Therefore,

The number of sp$^2$ hybridised carbon atoms in compound $X$ is $\boxed{8}.$

Since the reaction scheme is missing, this conclusion is based on the provided stored answer rather than an independent structural derivation.