\magnification=1440 \let\mathbb=\bf \def\E{\widetilde{\cal E}}
\hsize=12cm \hoffset=-5mm \def\alert#1{#1}

In the 2-dimensional case, $e_2(\varphi)$ can be computed as follows
(at least when $\varphi\in\E(\omega)$ has analytic singularities). 
Let $\gamma=\nu(\varphi,0)$ and $\mu:\widetilde\Omega\to \Omega$
be the blow-up of $\Omega$ at point $0$. Take local coordinates $(w_1,w_2)$
on $\widetilde\Omega$ so that the exceptional divisor $E$ can be
written $w_1=0$. Then 
\alert{$$\widetilde\varphi(w)=\varphi\circ\mu(w)-\gamma\log|w_1|$$}
is psh with generic Lelong numbers equal to $0$
along $E$, and therefore there are at most countably many points
$x_\ell\in E$ at which \hbox{$\gamma_\ell=\nu(\widetilde\varphi,x_\ell)>0$}.
Set $\Theta=dd^c\varphi$, $\widetilde\Theta=dd^c\widetilde\varphi=\mu^*\Theta
-\gamma[E]$. Since $E^2=-1$ in cohomology, we have
$\{\widetilde\Theta\}^2=\{\mu^*\Theta\}^2-\gamma^2$ in 
$H^2(E,{\mathbb R})$, hence
$$\int_{\{0\}}(dd^c\varphi)^2=\gamma^2+\int_E(dd^c\widetilde\varphi)^2.
\leqno(*)$$
If $\widetilde\varphi$ only has ordinary logarithmic poles at the $x_\ell$'s, 
then $\int_{E} (dd^c\widetilde\varphi)^2=\sum \gamma_\ell^2$, but in general the situation is more complicated. Let us blow-up any of the points $x_\ell$ and repeat the process $k$ times (we set $\ell=\ell_1$ in what follows, as this was the 
first step, and at step $k=0$ we omit any indices as $0$ is the only point
we have to blow-up to start with).
We then get inductively $(k+1)$-iterated blow-ups depending on multi-indices
$\ell=(\ell_1,\ldots,\ell_k)=(\ell',\ell_k)$ with
$\ell'=(\ell_1,\ldots,\ell_{k-1})$,
$$
\mu_\ell:\widetilde\Omega_\ell\to \widetilde\Omega_{\ell'}, ~~~k\ge 1,~~
\mu_\emptyset=\mu:\widetilde\Omega_\emptyset=\widetilde\Omega\to\Omega,~~
\gamma_\emptyset=\gamma
$$
and exceptional divisors $E_\ell\subset\widetilde\Omega_\ell$ lying over points
$x_\ell\in E_{\ell'}\subset\widetilde\Omega_{\ell'}$, where
$$
\eqalign{
&\gamma_\ell = \nu(\widetilde\varphi_{\ell'},x_\ell)>0,\cr
&\widetilde\varphi_\ell(w)=
\widetilde\varphi_{\ell'}\circ\mu_\ell(w)-
\gamma_\ell\log|w_1^{(\ell)}|,\cr}
$$
and $w_1^{(\ell)}=0$ is an equation for the exceptional divisor $E_\ell$
in the relevant coordinate chart. Formula $(*)$ implies
$$
e_2(\varphi)\ge\sum_{k=0}^{+\infty}~\sum_{\ell\in{\mathbb N}^k}~
\gamma_\ell^2\leqno(**)
$$
with equality when $\varphi$ has an analytic singularity at $0$.
We conjecture that $(**)$ is always an equality whenever $\varphi\in\E(\Omega)$.
This would imply the Guedj-Rashkovskii conjecture already mentioned.
Notice that the currents $\Theta_\ell=dd^c\widetilde\varphi_\ell$
satisfy inductively $\Theta_\ell=\mu_\ell^*\Theta_{\ell'}-\gamma_\ell[E_\ell]$, hence the cohomology class of $\Theta_\ell$ restricted to $E_\ell$ is equal
to $\gamma_\ell$ times the fundamental generator of $E_\ell$.
As a consequence we have
$$
\sum_{\ell_{k+1}\in{\mathbb N}} \gamma_{\ell,\ell_{k+1}}\le \gamma_\ell,
$$
in particular $\gamma_\ell=0$ for all $\ell\in{\mathbb N}^k$ if 
$\gamma=\nu(\varphi,0)=0$.
\end