for sample (D). For samples (A) to (C) the DW speed of the down/up DWs decrease down to the largest available negative $H_{x}$ field, with a saturation but not a clear minimum in the DW speed. This suggests that $H_{\mathrm{DMI}}$ in these samples is of the order or more than $+300$mT.

For sample (D), where the PMA (and therefore the $H_{\mathrm{DMI}}$ field) is reduced, the down/up DWs exhibit minimum speed for $H_{x}=\approx-180$mT, a value close to that found for the same sample in the creep regime. By taking $M_{s}=1 \times 10^{6}$A/m (measured by VSM-SQUID), $\mu_{0}H_{K}=0.7$T (measured by EHE) and $A=2.2 \times 10^{-11}$J/m [@Metaxas2007] the expression $H_{\mathrm{DMI}}=D/(\mu_{0}\Delta~M_{s})$ gives rise to a value of $D=1.27$mJ/m$^{2}$, with $\Delta=7.1$nm. Taking into account the larger thickness of the Co layer (1nm) in our Pt/Co/GdO$_x$ samples, this value scales reasonably well with the $D=2$mJ/m$^{2}$ value found for Pt/Co(0.6nm)/AlO$_x$ [@Pizzini2014; @Jue2015]. For samples (A) to (C) it is difficult to obtain a precise value of $D$ from the field-dependent measurements, where the minimum speed is not well defined. The value of $D$ for sample (A) may be derived from the in-plane field for which the DW chirality switches when driven by spin-polarised current ($H_{x}=\approx-280$mT). Using the values 