Difference-in-Differences time-variant control variable

I conduct a staggered difference-in-differences analysis across multiple groups i and multiple time periods t. Therefore, I fitted a fixed-effects model using the following equation:
$$
Y_{it}=beta_{o}Treatment_{it} + alpha_{i} + delta_{t} + varepsilon_{it}
$$

Treatment is dummy coded 1 for the first time period and all subsequent time periods after group i was treated, and coded 0 otherwise. Now I would like to include a time-variant control variable C into my model:
$$
Y_{it}=beta_{o}Treatment_{it} + beta_{1}C_{it} + alpha_{i} + delta_{t} + varepsilon_{it}
$$

I’m not sure whether C causes Y or vice versa. However, I have reason to believe that C causes the Treatment. Does this qualify C to be included as a control variable?