  In this paper, we continue the study of the total domination game in graphs
introduced in [Graphs Combin. 31(5) (2015), 1453--1462], where the players
Dominator and Staller alternately select vertices of $G$. Each vertex chosen
must strictly increase the number of vertices totally dominated, where a vertex
totally dominates another vertex if they are neighbors. This process eventually
produces a total dominating set $S$ of $G$ in which every vertex is totally
dominated by a vertex in $S$. Dominator wishes to minimize the number of
vertices chosen, while Staller wishes to maximize it. The game total domination
number, $\gamma_{\rm tg}(G)$, of $G$ is the number of vertices chosen when
Dominator starts the game and both players play optimally. Henning, Klav\v{z}ar
and Rall [Combinatorica, to appear] posted the $\frac{3}{4}$-Game Total
Domination Conjecture that states that if $G$ is a graph on $n$ vertices in
which every component contains at least three vertices, then $\gamma_{\rm
tg}(G) \le \frac{3}{4}n$. In this paper, we prove this conjecture over the
class of graphs $G$ that satisfy both the condition that the degree sum of
adjacent vertices in $G$ is at least $4$ and the condition that no two vertices
of degree $1$ are at distance $4$ apart in $G$. In particular, we prove that by
adopting a greedy strategy, Dominator can complete the total domination game
played in a graph with minimum degree at least $2$ in at most $3n/4$ moves.
