Let $\mathcal{H}$ be the separable Hilbert space associated to some quantum system, and let $\langle\cdot,\cdot\rangle :\mathcal{H}\times\mathcal{H}\rightarrow\mathbb{C}$ denote it's inner product. The projectivisation $\mathbb{P}\mathcal{H}$ of $\mathcal{H}$ then corresponds to the set of pure states of the system. The transition probability between two states $\Psi,\Phi\in\mathbb{P}\mathcal{H}$ is given by $$ T(\Psi,\Phi):=|\langle\psi,\phi\rangle| $$ where $\psi$ and $\phi$ are arbitrary unit vectors in the rays $\Psi$ and $\Phi$ respectively. In almost any elementary introduction to quantum mechanics, a symmetry is defined as a bijective map $s:\mathbb{P}\mathcal{H}\rightarrow\mathbb{P}\mathcal{H}$ which preserves transition probabilities i.e. $$ T(s\Psi,s\Phi)=T(\Psi,\Phi). $$ This is the definition of a symmetry for which Wigner's Theorem is relevant. The intuition behind it is clear, since a symmetry must preserve the physically relevant structures (i.e. probability amplitudes) of the theory. However, transition probabilities are only a subset of these structures.
More generally, each physical observable is identified with a self-adjoint operator $O:\mathcal{H}\rightarrow\mathcal{H}$. Denote the set of such observables by $\mathcal{O}$. Then the Born rule states that if a system is in the state $\Psi\in\mathbb{P}\mathcal{H}$, the probability that a measurement of the observable $O$ will produce a value within the Borel-measurable subset $U\subset \mathbb{R}$ is given by \begin{equation} \textrm{Prob}_{\Psi,O}(U):=\langle\psi,P_{O}(U)\psi\rangle, \end{equation} where $P_{O}:\mathcal{H}\rightarrow\mathcal{H}$ denotes the projection-valued-measure associated to the self-adjoint operator $O$ by the spectral theorem. This motivates the definition of a quantum symmetry as a pair of bijective maps, $s_{1}:\mathbb{P}\mathcal{H}\rightarrow\mathbb{P}\mathcal{H}$ and $s_{2}:\mathcal{O}\rightarrow\mathcal{O}$, which preserve the above probability measure. That is, for all $O\in\mathcal{O}$ and $\Psi\in\mathbb{P}\mathcal{H}$ \begin{equation} \textrm{Prob}_{\Psi,O}=\textrm{Prob}_{s_{1}(\Psi),s_{2}(O)}. \end{equation}
My question is whether these two definitions of a symmetry transformation can be shown to be equivalent in some sense, and if so, whether one must impose any constraints on the maps $s_{1}$ and $s_{2}$ to show this. In other words, does the preservation of transition probabilities guarantee the preservation of Born probabilities?