  In 2022, Gezmis and Pellarin introduced and studied the concept of trivial
multiple zeta values, along with a map from the vector space spanned by these
values to the vector space spanned by Thakur's multiple zeta values. Their
construction allows us to generate some linear relations among the latter
values using the former. In our work, we determine the structure of the kernel
of the aforementioned map. As a consequence, we give an answer to a conjecture
proposed by Gezmis and Pellarin regarding the injectivity of this specific map.
