Let the $7$ Teams be $A,B,C,D,E,F,G$ and so each team plays total 6 matches.
Suppose, Team $A$ wins over $E,F,G$ and draws with $B,C,D$ hence total points scored by Team A = $9$ points
Now, Team $B$ wins over $E,F,G$ and draws with $A,C,D$ hence total points scored by Team B = $9$ points
Similarly, happens for next two teams $C$ and $D$ .
Hence, Finalized scores are =>
A = 9
B = 9
C = 9
D = 9
E = ? (Less than or equal to 4)
F = ? ("...")
G = ? ("...")
Given that the order among the teams with the same total are determined by a whimsical tournament referee.
So, He/She can order the top $3$ teams like $ABC$,$ABD$,$BCD$,$ACD$ .......
But, Question says " team must earn in order to be guaranteed a place in the next round "
Hence, Not to depend on that whimsical referee, the minimum total number of points a team must earn in order to be guaranteed a place in the next round = $9+1$ = $10$ points