  In [1], a most general higher curvature non-local gravity action was derived
that admits a particular $R^2$-like inflationary solution predicting the
spectral index of primordial scalar perturbations $n_s(N)\approx
1-\frac{2}{N}$, where $N$ is the number of e-folds before the end of inflation,
$N\gg 1$, any value of the tensor-to-scalar ratio $r(N)<0.036$ and the tensor
tilt $n_t(N)$ violating the $r= -8n_t$ condition. In this paper, we compute
scalar primordial non-Gaussianities (PNGs) in this theory and effectively
demonstrate that higher curvature non-local terms lead to reduced bispectrum
$f_{\rm NL}\left( k_1,\,k_2,\,k_3 \right)$ mimicking several classes of scalar
field models of inflation known in the literature. We obtain $\vert f_{\rm
NL}\vert \sim O(1-10)$ in the equilateral, orthogonal, and squeezed limits and
the running of these PNGs measured by the quantity $\vert\frac{d\ln f_{\rm
NL}}{d\ln k}\vert\lesssim 1$. Such PNGs are sufficiently large to be measurable
by future CMB and Large Scale Structure observations, thus providing a
possibility to probe the nature of quantum gravity. Furthermore, we demonstrate
that the $R^2$-like inflation in non-local modification of gravity brings
non-trivial predictions which go beyond the current status of effective field
theories (EFTs) of single field, quasi-single field and multiple field
inflation. A distinguishable feature of non-local $R^2$-like inflation compared
to local EFTs is that we can have running of PNGs at least an order of
magnitude higher. In summary, through our generalized non-local $R^2$-like
inflation, we obtain a robust geometric framework of inflation that can explain
any detection of observable quantities related to scalar PNGs.
