ESG and (stock) beta

Through Capital Asset Pricing Model (CAPM) equilibrium the concept of beta has been introduced in order to decompose the inherent risk of a given security to systematic (or market) and firm-specific components. 
\(R_p\left(t\right)=\alpha_p+\beta R_M\left(t\right)+e_p\left(t\right)\) 
\(\sigma_p^2=\beta_p^2\sigma_M^2+\sigma_p^2\left(e\right)\)

Critical to observe that \(\beta\) influences both return and risk. By definition beta is a pure statistical construct regressed from historical data to measure the sensitivity of a given security to market which is evident from this definition

\(\beta=\frac{COV\left(r_{security},r_{Market}\right)}{\sigma^2_M}\)
While considering ESG factors, it seems although Environmental and Social risks are not firm-specific risks, Governance risks partially can be factored into the firm-specific risk component (\(\sigma_p\left(e\right)\)) -

\(\sigma^2_p=w^2_{fs}\sigma^2_{fs}+w^2_{gov}\sigma^2_{gov}+2\cdot w_{fs}\cdot w_{gov}\cdot COV\left(fs,gov\right)\)