Ramified and Unramified Motivic Multiple $t$-, $T$- and $S$-Values

In this paper we shall consider a few variants of the motivic multiple zeta values of level two by restricting the summation indices in the definition of multiple zeta values to some fixed parity patterns. These include Hoffman's multiple $t$-values, Kaneko and Tsumura's multiple $T$-values, and the multiple $S$-values studied previously by the authors. By applying Brown and Glanois's descent theory on the motivic versions of these values we shall derive some criterion for when these values are ramified and unramified. Assuming Grothendieck's period conjecture, our results partially confirms a conjecture of Kaneko and Tsumura about when multiple $T$-values can be expressed as a rational linear combination of multiple zeta values (i.e., unramified) if their depth is less than four. Similar results are obtained for motivic multiple $S$-values. Further, we are able to generalize a result of Charlton to more families of unramified multiple $t$-values with unit components (i.e. components equal to 1). We propose some more unsolved problems at the end of the paper.