This should follow from the following Virial-type computation.
For convenience we assume all particles have the same mass; this is not essential.
Let $x_i$ denote the position vector of the $i$th particle, then Newton's law of universal gravitation, suitably normalized, gives
$$ \ddot{x_i} = (d-2) \sum_{j \neq i} \frac{ x_j - x_i}{|x_j - x_i|^d } $$
This gives
$$ x_i \cdot \ddot{x_i} = (d-2) \sum_{j \neq i} \frac{ x_j \cdot x_i - |x_i|^2 }{|x_j - x_i|^d} $$
summing over all particles we get
$$ \sum_{i} x_i \cdot \ddot{x_i} = (d-2) \sum_{i,j, i\neq j} \frac{ x_j \cdot x_i - |x_i|^2 }{|x_j - x_i|^d} = - \frac{d-2}{2} \sum_{i,j, i\neq j} |x_i - x_j|^{d-2} \tag{1}$$$$ \sum_{i} x_i \cdot \ddot{x_i} = (d-2) \sum_{i,j, i\neq j} \frac{ x_j \cdot x_i - |x_i|^2 }{|x_j - x_i|^d} = -\frac{d-2}{2} \sum_{i,j, i\neq j} |x_i - x_j|^{2-d} \tag{1}$$
Conservation of energy, on the other hand, states that
$$ - \sum_{i,j, i\neq j} |x_i - x_j|^{d-2} + \sum_i |\dot{x_i}|^2 = C < 0 \tag{2}$$$$ - \sum_{i,j, i\neq j} |x_i - x_j|^{2-d} + \sum_i |\dot{x_i}|^2 = C < 0 \tag{2}$$
(by negative energy assumption). Summing (1) and (2) and using that $d \geq 4$ so that $(d-2)/2 \geq 1$ we get
$$ \frac{d^2}{dt^2} \sum_{i} |x_i|^2 < 0 $$
Which shows that $\sum |x_i|^2$ must impact $0$ at some finite time (either future or past).
(Note: by "dropping particles at infinity" and using time-symmetric property of Newtonian mechanics we see that there exists trajectory where there is only one time of impact in the past, and none in the future.)