Answer
The period remains the same.
Work Step by Step
We can write an expression for the period:
$T = 2\pi \sqrt{\frac{I}{mgx}}$
$T = 2\pi \sqrt{\frac{\frac{1}{12}mL^2+mx^2}{mgx}}$
$T = 2\pi \sqrt{\frac{\frac{1}{12}L^2+x^2}{gx}}$
We can see that the period does not depend on the mass.
Therefore, even if the mass increases, then the period remains the same.
