Per Wikipedia, the Ebers-Moll equation for \$I_C\$ is
$$I_C=I_S[(e^{\frac{V_{BE}}{V_T}}-e^{\frac{V_{BC}}{V_T}})-\frac{1}{\beta_R}(e^{\frac{V_{BC}}{V_T}}-1)]$$
When the collector and emitter are at the same potential, \$V_{BE}=V_{BC}\$ so the Ebers-Moll equation for \$I_C\$ reduces to
$$I_C=-\frac{I_S}{\beta_R}(e^{\frac{V_{BC}}{V_T}}-1)$$
By similar reasoning, when collector and emitter are at the same potential, the equation for \$I_E\$ is reduced to
$$I_E=\frac{I_S}{\beta_F}(e^{\frac{V_{BE}}{V_T}}-1)$$
The ratio between \$I_C\$ and \$I_E\$ is then found to be
$$\frac{I_C}{I_E}=-\frac{\beta_F}{\beta_R}$$
where \$I_C\$ and \$I_E\$ are taken in their usual direction, or
$$\frac{I_C}{I_E}=\frac{\beta_F}{\beta_R}$$
where \$I_C\$ and \$I_E\$ are both taken as currents flowing out of the device.
That is, the base-collector and base-emitter junctions act like a pair of forward biased diodes in the circuit in question with the base-collector diode conducting a larger current, approximately by the proportion \$\frac{\beta_F}{\beta_R}\$.
When simulated with Falsad, using a \$\beta\$ value of 100, the \$\frac{I_C}{I_E}\$ ratio works out to almost exactly 100.
When simulated with CircuitLab, using a 2N2222 model, with a \$\beta_F\$ of 100 and a \$\beta_R\$ of 5, the \$\frac{I_C}{I_E}\$ ratio works out to 114.8/81.36 = 1.411 which is quite far from the prediction of 20. This suggests that CircuitLab uses a model other than Ebers-Moll. By modifying the values of Rc and Re in the CircuitLab model for the transistor, it becomes clear that the division of current between emitter and collector is highly dependent on the internal resistances of the transistor, and not so much on the ratio of \$\beta_F\$ to \$\beta_R\$. I thus suspect that the CircuitLab result is closer to reality than the Falstad result.