Variance of the truncated normal distribution (truncated from below) increases in $\sigma$? I'm wondering whether the variance of the truncated normal distribution increases in $\sigma$ (which seems to hold numerically), where the untruncated normal distribution is $N(\mu,\sigma^2)$ and the truncated normal distribution is truncated from zero. 
I found that there were some discussions on the relationship between the mean of the truncated normal distribution and $\mu$ (Is the mean of the truncated normal distribution monotone in $\mu$?), and the relationship between the mean of the truncated normal distribution and $\sigma$ (Effect of variance on truncated normal) but couldn't find any discussion on the relationship between the variance (or standard deviation) of the truncated normal distribution and $\sigma$.
The variance of the truncated normal distribution (truncated from below) is:
$Var(X|X>0)=\sigma^2 \left[1+\frac{\left(-\frac{\mu}{\sigma}\right) \phi\left(-\frac{\mu}{\sigma}\right)}{1-\Phi\left(-\frac{\mu}{\sigma}\right)} -\left( \frac{\phi\left(-\frac{\mu}{\sigma}\right)}{1-\Phi\left(-\frac{\mu}{\sigma}\right)} \right)^2\right]$
$\Phi,\phi$ are cdf and pdf of the standard normal distribution. 
Is there any proved claim that $Var(X|X>0)$ increases in $\sigma$? Or can we prove it? Any information or insight would greatly help.
 A: A straightforward (computer) symbolic integration shows that the variance of the truncated distribution for arbitrary $\mu$ gives: 
$$\sigma  \left(-\frac{2 \sigma  e^{-\frac{\mu ^2}{\sigma ^2}}}{\pi 
   \left(\text{erf}\left(\frac{\mu }{\sqrt{2} \sigma
   }\right)+1\right)^2}+\frac{\sqrt{\frac{2}{\pi }} \mu  e^{-\frac{\mu ^2}{2 \sigma
   ^2}}}{\text{erfc}\left(\frac{\mu }{\sqrt{2} \sigma }\right)-2}+\sigma \right).$$
Here's a graph (for fixed $\mu = 1$):

Here's the derivative of the variance with respect to $\sigma$:
$$-\frac{4 \sqrt{2} \mu  e^{-\frac{3 \mu ^2}{2 \sigma ^2}}}{\pi ^{3/2}
   \left(\text{erf}\left(\frac{\mu }{\sqrt{2} \sigma }\right)+1\right)^3}-\frac{2
   e^{-\frac{\mu ^2}{\sigma ^2}} \left(3 \mu ^2+2 \sigma ^2\right)}{\pi  \sigma 
   \left(\text{erf}\left(\frac{\mu }{\sqrt{2} \sigma
   }\right)+1\right)^2}+\frac{\sqrt{\frac{2}{\pi }} \mu  e^{-\frac{\mu ^2}{2 \sigma ^2}}
   \left(\mu ^2+\sigma ^2\right)}{\sigma ^2 \left(\text{erfc}\left(\frac{\mu }{\sqrt{2}
   \sigma }\right)-2\right)}+2 \sigma$$
Here's a graph of the variance with respect to $\sigma$ and $\mu$:  always monotonic:

