No, it is not true;
Consider the matrix

$$A=
\begin{pmatrix}
\frac{9}{2} & \frac{9}{20} & \frac{21}{20} & -\frac{3}{2}\\
-\frac{79}{11} & -\frac{3}{110} & \frac{23}{110} & \frac{31}{11}\\
\frac{6}{11} & \frac{21}{55} & \frac{114}{55} & \frac{6}{11}\\
\frac{16}{11} & \frac{12}{55} & \frac{128}{55} & \frac{5}{11}
\end{pmatrix}
$$
which has eigenvalues 0,1,2,3 (so it is positive semi-definite, but not definite.)
The four principal minors are
$$\frac{27}{22},\frac{189}{110},\frac{153}{55},\frac{36}{11}$$
sorted in increasing order. This should give a definite negative answer to your question.