Thanks a lot for your help. So it looks like individual $\hat{H}(t; z)$ is based on the non-parametric device, $\hat{H}_0(t)$, as well as the results from Cox regression $e^{\sum \beta z}$.

Btw, I seem to have managed to show that $\exists!\tau\in\mathbb{R}^+:f_1(\tau)=f_2(\tau)$. I think solution to the question itself will follows

@whuber According to the data from Lourens et al, ML-based estimation of bimodal normal mixture shows systematic biases, including negative biases for $\mu_1$ and $\sigma$, and positive biases for $\mu_2$ etc., which are worse with larger true $OVL$. We are not working on normal mixtures. Our work involves an MLE process of a Bayesian hierarchical model that involves the estimation of multiple parameters, including a log-normally distributed parameter of interest which consists of a mixture as described in the question. We want to describe numerically how $OVL$ relate with biases in our case.

@whuber Lourens, S., Zhang, Y., Long, J. D., & Paulsen, J. S. (2013). Bias in Estimation of a Mixture of Normal Distributions. Journal of biometrics and biostatistics, 4, 1000179. Lourens et al in turn cited this: Henry F. Inman & Edwin L. Bradley Jr (1989) The overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities, Communications in Statistics - Theory and Methods, 18:10, 3851-3874

@whuber $OVL$ that I quoted in my question is something called the overlapping coefficient, whose converse (i.e. $1-OVL$) has been used as a disparity index to associate with the estimation quality of a mixture of two normal distributions (I will put relevant papers in the next comment). These work and those suggested by user2974951 are for normal but not log-normal distributions. $f_1$ (the red curve) and $f_2$ (the green curve) is the pdf of the log-normal distribution. I am guessing I need and I am trying to solve for the intersections first, but my algebraic skill is not that good...

Thanks. I have seen these, but they refer to normal distributions and also do not have the proportion parameter defined in my question.