Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d$. Since $\dim(R/P)+\text{ht}(P)\le d=\dim(R/\mathfrak m)+\text{ht}(\mathfrak m)$ holds for every prime ideal $P$ of $R$, so $$\sup \{ \dim(R/P)+\text{ht}(P): P\in \text{Spec}(R) \}=d$$
Now in general, there can exist prime ideal(s) $P$ such that $\dim(R/P)+\text{ht}(P)<d$.
So my question is: Is there a good lower bound for $$\inf \{ \dim(R/P)+\text{ht}(P): P\in \text{Spec}(R) \}$$ in terms of $d$ ?
In a more concrete direction, is it atleast true that $$\inf \{ \dim(R/P)+\text{ht}(P): P\in \text{Spec}(R) \}=\inf \{ \dim(R/P): P\in \text{Min}(R) \}$$?
Note that if $R$ is Cohen-Macaulay (or more generally, quasi-unmixed), then $\dim(R/P)+\text{ht}(P)=d$ for every prime ideal $P$ of $R$, so I am mainly asking for the non Cohen-Macaulay case.
Thanks in advance.