In adaptive video streaming, optimizing the selection of representations for the encoding bitrate ladder has a significant impact on the quality and economics of media delivery. An efficient way to select representations for the bitrate ladder of a given clip is to consider the Satisfied User Ratio (SUR) of the perceived quality of consecutive representations. This ensures that only representations with one Just Noticeable Difference (JND) are encoded and streamed by avoiding encoding similar-quality representations. VMAF (Video Multi-method Assessment Fusion) presently stands as the most commonly utilized quality metric for constructing bitrate ladders. Hence, the precise determination of JND-optimal encoding step-sizes for the VMAF proxy holds paramount importance; nevertheless, this task is intricate and can present considerable challenges. In this paper, we evaluate the effectiveness of different Video Quality Metrics (VQMs) in predicting SUR for the VMAF proxy to better capture content-specific characteristics. Our experimental results provide evidence that incorporating VQMs can improve the precision of the SUR prediction for the VMAF proxy. Compared to a state-of-the-art approach that utilizes video complexity metrics, our proposed approach, which incorporates two quality metrics—specifically, VMAF and SSIM calculated at an optimized quantization parameter (QP)—achieves a substantially reduced Mean Absolute Error (MAE) of 1.67. In contrast, the state-of-the-art approach yields an MAE of 2.01. Hence, we recommend using the above quality metrics to improve the accuracy of the SUR prediction for the VMAF proxy.