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Let $G$ be a $3$-connected triangulated disc of order $n$ with the boundary cycle $C$ of the outer face of $G$. Tokunaga (2013) conjectured that $G$ has a dominating set of cardinality at most $\frac{1}{4}(n+2)$. This conjecture is proved in Tokunaga (2020) for $G-C$ being a tree. In this paper we prove the above conjecture for $G-C$ being a unicyclic graph. We also deduce some bounds for the double domination number, total domination number and double total domination number in triangulated discs.